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Int. J. Data Envelopment Analysis (ISSN 2345-458X)

Vol.5, No.2, Year 2017 Article ID IJDEA-00422, 24 pages

Research Article

Resource allocation based on DEA for distance

improvement to MPSS points considering

environmental factors

A. Mottaghi1*

, R.Ezzati2, E.Khorram

1

(1)

Faculty of Mathematics and Computer Science, Amirkabir University of

Technology, 424 Hafez Avenue, 15914 Tehran, Iran. (2)

Department of Mathematics, Islamic Azad University, Karaj Branch, Karaj,

Iran.

Received January 7, 2017, Accepted May 28, 2107

Abstract

This paper proposes a new resource allocation model which is based on data envelopment

analysis (DEA) and concerns systems with several homogeneous units operating under

supervision of a central unit. The previous studies in DEA literature deal with

reallocating/allocating organizational resource to improve performance or maximize the total

amount of outputs produced by individual units. In those researches, it is assumed that all data

are discretionary. Resource allocation problem has a multiple criteria nature; thus to solve it,

many intervening factors should be regarded. This paper not only develops resource allocation

plan for systems with both discretionary and non discretionary data in their inputs, but also

considers environmental factors as well. In addition, the overall distance from the decision

making units (DMUs) to their most productive scale size (MPSS) points is taken into account

and is minimized in this method. To find the best allocation plan, this paper applies multiple

objective programming (MOLP). Numerical examples are employed to illustrate the

application of this approach on real data.

Keywords: Resource allocation, DEA, MOLP, MPSS point, Undesirable outputs,

Discretionary inputs.

*. Corresponding author, E-mail: [email protected]

International Journal of Data Envelopment Analysis Science and Research Branch (IAU)

A. Mottaghi, et al /IJDEA Vol.5, No.2, (2017) 1207-1230

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1. Introduction Resource allocation is an important issue

in corporation management. In real life,

the resources are always limited, so how

to allocate them plays a pivotal role in

determining a corporation's growth. As a

result, resource allocation has been an

interesting topic for both corporations and

researchers.

Data envelopment analysis (DEA) is a

methodology for measuring the relative

efficiencies of a set of decision making

units (DMUs) that use multiple inputs to

produce multiple outputs. It was first

introduced by Charnes et al.[1].

DEA is a valid method from both

theoretical and empirical sides and it is

applicable to management process,

performance estimation and behavior

analysis. DEA models bring dramatic

chances when used as a method for

analyzing or solving resource allocation

problems. DEA assumes homogeneity

among DMUs in terms of the nature of

the operations they perform, the measures

of their efficiency and the conditions

under which they operate. When DMUs

are not homogeneous, the efficiency

scores may reflect the underlying

differences in environments rather than

inefficiencies. One strategy to tackle this

issue is to separate DMUs into

homogenous groups [2].

In recent years, DEA has been wildly

used by managerial researchers to study

how to better allocate resource. The

research about resource allocation using

DEA may be classified into two

categories. One category assumes the

efficiency of DMUs to be constant [3, 4,

5, 6] while the other assumes the

efficiency of DMUs to be changeable [4,

7, 8, 9]. In this paper, the latter category

where the efficiency of DMUs are

changeable is studied. Golany and Tamir

[10] have proposed a DEA- based output

oriented resource allocation model which

includes constraints that imposes upper

bounds on the total input consumption of

the target points. Care must be taken since

constraining the model too much could

result in infeasibility. Since this flexible

model aims at trading off efficiency,

effectiveness and equality in resource

consumption, it is rather complex.

Athanassopoulos [11] has presented a

DEA-based goal programming model

(GODEA) for centralized planning.

Global targets for the total consumption

of each input and the total production of

each output are approximated by solving

a series of independent DEA models for

each input and each output. These goals

are usually not simultaneously attainable

but deviations from them can be

minimized, resulting in an appropriate

resource allocation. Although existing

DMUs are jointly projected, the

individual units are not necessarily

projected onto the efficient frontier.

Athanassopoulos[12] has also proposed

another goal programming model yet this

time not based on the envelopment DEA

formulation but on the multiplier form.

The global targets for each input and

output are computed as in GODEA and

the deviation from them constitute the

highest priority element in the objective

function. The second priority is to

maximize operational efficiency onto the

part of the computed input and output

quantities. In the third place, the objective

function also tries to minimize the

inequality of resource consumption

among units. Beasley [7] presents a non-

linear resource allocation model to jointly

compute inputs and outputs for each

DMU for the next period with the

objective of maximizing the average

efficiency. The approach is based on a

non-linear ratio form formulation, (which

can lead to alternative optima) and

requires explicit upper limits on the total

amount of every input and of every

output. Nevertheless, there is no

guarantee that the projected points lie on

the efficient frontier.

Lozano and Villa [13] have proposed two

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new models for centralized resource

allocation. The first model seeks radial

reductions of the total consumption of

every input while the second model seeks

separate reduction levels for the total

amount of each input according to a

preference structure. The two key features

of the proposed models are their

simplicity and the fact that both of them

project all DMUs onto the efficient

frontier. The radial model jointly projects

each of the existing DMUs onto the

pareto efficiency frontier. Despite a high

probability that for every input the total

consumption is necessarily lower than the

sum of the input levels of the

independently input-oriented projected

units, this cannot be proved.

Korhonen and Syrjanen [14] have

developed an interactive approach based

on DEA and MOLP, which is capable of

dealing with several assumptions imposed

by DM.

Bi et al. [15] propose a methodology for

resource allocation and target setting

based on DEA for parallel production

systems. It deals with any organization

with several production units which have

parallel production lines.

Crisp input and output data are

fundamentally indispensable in

conventional DEA. However, the

observed values of the input and output

data in real word problems are sometimes

imprecise or vague. Many Researchers

have proposed various method for dealing

with the imprecise and ambiguous data in

DEA [16].

Although in most available resource

allocation models the key point is to

maximize the total amount of outputs

produced by individual units, they are not

concerned about the improvement in the

efficiency score of individual units or

even the efficiency of the whole system.

Nasrabadi et al. [17] present a model to

investigate the resource allocation

problem based on efficiency

improvement. In their model, the

parameters used are not necessarily

unique in the case of alternative optimal

solution. However, each optimal solution

can be applied in this model to achieve

performance improvement. This is a

shortcoming, of their model, since finding

all alternative optimal solutions and

solving the model for each one seems

unreasonable.

The two factors which often have some

relation with each other and play

important roles in resource allocation

models are economic and environmental

factors. Economic factors usually refer to

the desirable outputs generated in the

production process, such as profit.

Environmental factors usually refer to the

undesirable outputs such as smoke

pollution and waste. Jie and et al. [18]

have proposed some new DEA models

which consider both economic and

environmental factors in the allocation of

a given resource. Research on undesirable

inputs and outputs has also been actively

pursued by means of DEA.

Environmental factors are very important

in resource allocation, few papers have

provided methods in this regard. For the

first time, the current paper not only

considers environmental and economic

factors in resource allocation in the light

of DEA, but also it deals with

discretionary and non discretionary data,

simultaneously.

In the previous literature it was assumed

that all inputs and outputs can be varied at

the discretion of management or other

users. These may be called "discretionary

variables" . "Non discretionary variables",

which are not subject to management

control, may also need to be considered.

This paper aims to involve this variable as

well.

The assumptions that concern the units'

ability to change their input-output mix


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and efficiency are clearly some of the key

factors affecting the results of resource

allocation. Although many valuable ideas

have been proposed concerning these

assumptions, the DMUs ability to change

their input-output mix and efficiency has

not been discussed thoroughly in the

literature. In addition, the multiple criteria

nature of the resource allocation problem

has drawn only limited attention.

Resource allocation, which is a decision

problem in which the decision maker

(DM) allocates future available resources

to a number of similar units, has a

multiple criteria nature. This paper aims

to decrease the distance from units to

their MPSS points through the application

of DEA and MOLP.

This study assumes that several

homogeneous units operating under the

supervision of a central unit such that

these units consume some desirable and

undesirable inputs to produce very

desirable and undesirable outputs.

Moreover, a few of the inputs can be of

non discretionary type. Thus resource

allocation here includes both desirable

and undesirable inputs and outputs. The

literature in this area may be classified

into two categories: direct approaches and

indirect ones. Direct approaches are based

on the work of Fare et al. [19], which

replaced the strong disposability of

outputs by the assumption that outputs are

weakly disposable, and have then been

largely extended. Indirect approaches

may be further classified into two groups.

The first group deals with the undesirable

outputs as inputs for processing [20, 21];

while the second ones transform data for

undesirable outputs and then use the

traditional efficiency model for their

evaluation [22]. The above mentioned

indirect approaches are used to handle the

undesirable inputs and outputs.

The proposed model intends to address

the arisen problems when resources are

allocated to various DMUs such that the

distance between DMUs and their MPSSs

is minimized. On the other hand, the

current method improves efficiency and

return to scale of DMUs. In the new

method not only the economic and

environmental factors are considered but

also it is assumed that some data are non

discretionary.

The rest of this paper is organized as

follows:

Some theoretical aspects of MOLP, DEA

and MPSS points are discussed in section

2. In section 3, the proposed approach for

resource allocation is described. Section 4

shows the application of this method in

agriculture and petroleum industry. Next

in section 5, the proofs of the theorems

from section 3 are stated. Finally,

conclusions are presented in section 6.

2. Preliminary Considerations

2.1. Envelopment DEA technology A system with n DMUs, each consuming

m inputs and producing p outputs is

considered. Assume a production process

in this system where desirable and

undesirable inputs are consumed and

desirable and undesirable outputs are

jointly produced. In addition, some of the

inputs probability are discretionary.

Let xk = [(xik)i∈DDI, (xjk)j∈DUI,

(xlk)l∈NDDI, (xmk)m∈NDUI]

for k = 1, . . . , n, denotes the vector of

inputs of DMUk; (k = 1, . . . , n) which

DDI, DUI, NDDI, NDUI are the index

sets which include indexes of

discretionary desirable inputs,

discretionary undesirable inputs, non

discretionary desirable inputs and non

discretionary undesirable inputs,

respectively. Also, denote the vector of

outputs of DMUk by

yk = [(ytk)t∈DO, (ysk)s∈UO] where DO

and UO are the index sets of desirable

outputs and undesirable outputs,

respectively. The production technology

can be described as:

P = {(xDDI, xDUI, xNDDI, xNDUI, yDO, yUO)|

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(xDDI, xDUI, xNDDI, xNDUI) can produce (yDO, yUO)}

In order to reasonably model a production

technology that consumes both desirable

and undesirable inputs to produce

desirable and undesirable outputs, the

assumptions proposed by Liu et al are

adopted [20]. Under these assumptions,

the extended strong disposability can

formally be stated as:

If (xDDI, xDUI, xNDDI, xNDUI, yDO, yUO) ∈ P,

then for every (wDDI, wDUI, wNDDI, wNDUI, zDO, zUO) such that

wDDI ≥ xDDI, wDUI ≤ xDUI, wNDDI

≥ xNDDI, wNDUI ≤ xNDUI and zDO ≤ yDO zUO

≥ yUO,

we will have:

(wDDI, wDUI, wNDDI, wNDUI, zDO, zUO) ∈ P. Hence, the corresponding production

possibility set (PPS) is as follows:

P =

{

(xDDI, xDUI, xNDDI, xNDUI, yDO, yUO)

|

|

|

|xDDI ≥ ∑ n

j=1 λjxjDDI

xDUI ≤ ∑ nj=1 λjxj

DUI

yDO ≤ ∑ nj=1 λjyj

DO

yUO ≥ ∑ nj=1 λjyj

UO

∑ nj=1 λj = 1, λj ≥ 0; j = 1, . . . , n. }

2.2. Most Productive Scale Size

(MPSS) points Definition 1.( Banker's definition [23]):

(xo, yo) is MPSS if and only if for every

(αxo, βyo) ∈ P we have α ≥ β.

Jahanshahloo and Khodabakhshi [24]

used the input-output orientation model

for determining MPSS points

corresponding to DMUs. For this aim,

they solved the following model:

Max φo − θo

s. t φoyo ≤ ∑ nj=1 λjyj

θoxo ≥ ∑ nj=1 λjxj

∑ nj=1 λj = 1 (1 − I)

λj ≥ 0; j = 1, . . . , n.

Theorem 1. DMUo is MPSS if and only if

the two following conditions are satisfied:

(a) The optimal amount of objective

function is zero.

(b) The amounts of slacks in alternative

optimal solutions are zero.

Proof: Refer to Reference[24].

For determining MPSS points using

model (1 − I), the non- Archimedean

form is applied. The non- Archimedean

model is defined as follows:

Max φo − θo + ε(1. S− + 1. S+)

s. t Xλ + S− − θoxo = 0 (1 − II)

−Yλ + S+ + φoyo = 0

1. λ = 1λ ≥ 0, S− ≥ 0, S+ ≥ 0

This model is solved by a pre-emptive

approach such that at first the max

(φo − θo) is obtained without any

attention to slacks and then in the second

stage the slacks are maximized by fixing

φo∗ , θo

∗ amounts instead of φo, θo.

Therefore, in this approach it is not

necessary to devote any amount for ε. Remark 1. For DMUo with (xo, yo) input

and output combinations, figurative DMU

with (θ∗ xo - S−∗ , φ∗ yo+ S+∗) input and

output combinations is MPSS.

Also Khodabakhshi [25] estimates the

most productive scale size in stochastic

data envelopment analysis (DEA).To

estimate the most productive scale size


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with stochastic data, he develops the

input-output orientation model that was

introduced by Jahanshahloo and

Khodabakhshi [24] in classic DEA in

stochastic data envelopment analysis. The

deterministic equivalent of stochastic

model is obtained which is generally non-

linear, and it can be converted to a

quadratic problem.

In the following, we extend this model for

desirable and undesirable inputs and

outputs.

First, the following model is solved: Max φ1o − θ1o −φ2o + θ2o

s. t φ1oyro ≤ ∑ nj=1 λjyrj ; r ∈ DO

θ1oxio ≥ ∑ nj=1 λjxij ; i ∈ DDI

φ2oyro ≥ ∑ nj=1 λjyrj ; r ∈ UO

θ2oxio ≤ ∑ nj=1 λjxij ; i ∈ DUI

xio ≥ ∑ nj=1 λjxij ; i ∈ NDDI

xio ≤ ∑ nj=1 λjxij ; i ∈ NDUI

∑ nj=1 λj = 1

λj ≥ 0 j = 1, . . . , n.

Then the following model is solved:

Max Σi∈DDI 𝑠𝑖 + Σi∈DUI si + Σr∈DO 𝑡𝑟 + Σr∈UO 𝑡�� + Σi∈NDDI si + Σi∈NDUI si s. t φ1o

∗yro = ∑ nj=1 λjyrj − tr; r ∈ DO

θ1o∗xio = ∑ n

j=1 λjxij + si; i ∈ DDI

φ2o∗yro = ∑ n

j=1 λjyrj + 𝑡��; r ∈ UO

(2-II)

θ2o∗xio = ∑ n

j=1 λjxij − si; i ∈ DUI

xio = ∑ nj=1 λjxij + si; i ∈ NDDI

xio = ∑ nj=1 λjxij − si; i ∈ NDUI

∑ nj=1 λj = 1, λj ≥ 0; j = 1, . . . , n.

Similarly, it can be proven for DMUo with

(xoDDI, xo

DUI, xoNDDI, xo

NDUI, yoDO, yo

UO)

combination, figurative DMU with the

following form:

(θ1∗xoDDI − S∗, θ2

∗xoDUI + S∗, xo

NDDI −

S∗, xoNDUI + S∗, φ1

∗yoDO + t∗, φ2

∗yoUO − t∗)

is MPSS.

Here, for determining the distance

between DMUo and its MPSS, the

following formulation is applied:

ρo = Σi∈DDI (xio(1 − θ1o∗ ) + si

∗) +

Σi∈DUI (xio(θ2o∗ − 1) + si

∗) + Σr∈DO(yro(φ1o

∗ − 1) + tr∗) +

Σr∈UO(yro(1 − φ2o∗ ) + tr

∗) +

Σi∈NDDI si∗ + Σi∈NDUI si

∗.

2.3 Multi-objective programming Here, some fundamentals of MOP

problems and the weighted sum method

for solving them are reviewed, which will

be used throughout the remainder of this

paper.

The MOP problem can be presented as

follows:

max f(x) = (f1(x), f2(x), . . . , fp(x))

s. t.x ∈ χ

where χ is a feasible set of an

optimization problem (4) and fk: χ → ℝ

for k = 1, . . . , p are criteria or objective

functions. The fundamental importance of

efficiency (Pareto optimality) is based on

the observation that any x which is not

efficient cannot represent a most

preferred alternative for a DM, because

there exists at least one other feasible

solution x′ such that fk(x′) ≥ fk(x) for all

k = 1, . . . , p, where strict inequality holds

at least once, i.e., x′ should clearly be

preferred to x.

The multi-objective linear programming

(MOLP) problems are specified by linear

functions which are to be maximized

subject to a set of linear constraints. The

standard form of MOLP can be written as

follows:

(2–I)

(3)

(4)

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max f(x) = Cx

s. tx ∈ χ = {x ∈ ℝn|Ax ≤ b, x ≥ 0}

where C is an p × n objective function

matrix, A is m× n constraint matrix, b is

an m- vector of right hand side, and x is

an n-vector of decision variables.

Definition 2. (Ehrgott [26]) Let x∗ be a

feasible solution to the problem (5), if

there is no feasible solution x of (5) such

that (Cx∗)k ≤ (Cx)k for all k = 1,2, . . . , p

and (Cx∗)k < (Cx)k for at least one i,

then we say x∗ is a strongly efficient

solution of (5).

There are many methods for solving

MOLP problems. One of the non

scalarizing methods is weighted sum

method. In this method, an MOP problem

(5) can be solved (i.e. its efficient

solutions be found) by solving a single

objective problem of this type: max {λTCx | x ∈ χ}

(6)

where λk ∈ Λ and Λ = {λ ∈ ℝk | λi ≥ 0,

Σi=1k λi = 1}.

Theorem 2: x ∈ χ is efficient if and only

if there exists a λ ∈ Λ such thatx is a

maximized solution of max{λTCx | x ∈ χ}.

Proof: Refer to Reference [27].

3. Development of resource

allocation model In this section, a new model for the

resource allocation is proposed. Here

resource allocation means a decision

problem in which the decision maker

wishes to allocate extra resources as new

inputs and additional market demands as

new outputs for a set of homogeneous

units operating in a system to achieve

more small distances to MPSS points of

these units.

Assume that (xkDDI + Δxk

DDI, xkDUI +

ΔxkDUI, xk

NDDI, xkNDUI, yk

DO + ΔykDO, yk

UO +

ΔykUO) represents the activity vector of

DMUk after the planning period, in which

ΔxkDDI and Δxk

DUI denote the vector of

discretionary desirable input changes and

discretionary undesirable input changes of

DMUk, for k = 1, . . . , n respectively.

Also, ΔykDO and Δyk

UO denote the vector

of desirable output changes and

undesirable output changes of DMUk, for

k = 1, . . . , n.

Definition 3. The system containing

DMU1, . . . , DMUn has a smaller distance

to MPSS points, after the planning period

if (ρk

ρk′) > 1 for k: 1, . . . n where ρk and

ρk′ are the distance measure between

DMUk and its MPSS, before and after the

planning period, respectively.

It should be maximized

{ρ1

ρ1, . . . ,

ρk

ρk′, . . . ,

ρn

ρn′}, while maintaining

the feasibility of all units and imposing

the DM's constraints, to achieve a smaller

distance to MPSS points.

Max {ρ1

ρ1, . . . ,

ρk

ρk′, . . . ,

ρn

ρn′}

s. t (xkDDI + Δxk

DDI, xkDUI + Δxk

DUI,

xkNDDI, xk

NDUI, ykDO + Δyk

DO, ykUO + Δyk

UO) ∈ P

; k = 1, . . . , n

(Δx1DDI, Δx2

DDI, . . . , ΔxnDDI) ∈ ΔDDI

(Δx1DUI, Δx2

DUI, . . . , ΔxnDUI) ∈ ΔDUI

(Δy1DO, Δy2

DO, . . . , ΔynDO) ∈ ΔDo

(Δy1UO, Δx2

UO, . . . , ΔxnUO) ∈ ΔUO.

Model (7) has n objectives and is a

multiple objective problem. To solve this

model, the weighted sum method is used.

Therefore the following model is

obtained:

(5)

(7)


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Max ∑ nk=1 wk(

ρk

ρk′)

s. t ConstraintsofModel(7), (8)

where wks are positive weights

representing the DM’s preferences such

that ∑ nk=1 wk = 1.

Here, it is important to know, if a

decrease in the distance measure occurs

from all units to their MPSS points, the

system will surely have a smaller

distance, whereas, in general the converse

does not hold true. For considering this

fact in the new model, it was assumed

that the DM is not willing to have

deterioration in distance measure from

any units to its MPSS, which implies

ρk′ ≤ ρk for k = 1, . . . , n. Then the

resource allocation model is introduced as

follows:

Max ∑ nk=1 wk(

ρk

ρk′)

s. t ConstraintsofModel(7),

ρk′ ≤ ρk; k = 1, . . . , n

(9)

where sets ΔDDI, ΔDUI, ΔDO, ΔUO, refer to

the restrictions imposed on the vectors of

discretionary desirable and undesirable

input changes and desirable and

undesirable output changes, respectively.

Note that we can choose these sets

according to the DM's preferences. Here

we consider them as follows:

ΔDDI = {(Δxi1DDI, Δxi2

DDI, … , ΔxinDDI)|

αik ≤ ΔxikDDI ≤ βik ; k = 1,… , n

Σk=1n Δxik

DDI = ci ; i ∈ DDI}

ΔDUI = {(Δxi1DUI, Δxi2

DUI, . . . , ΔxinDUI)|

αik′ ≤ ΔxikDUI ≤ βik′ ; k = 1, . . . , n

, Σk=1n Δxik

DUI = ci ; i ∈ DUI}

ΔDO = {(Δyr1DO, Δyr2

DO, . . . , ΔyrnDO)|

γrk ≤ ΔyrkDO ≤ δrk ; k = 1, . . . , n

, Σk=1n Δyrk

DO = dr ; r ∈ Do}

ΔUO = {(Δyr1UO, Δyr2

UO, . . . , ΔyrnUO)|

γrk′ ≤ ΔyrkUO ≤ δrk′ ; k = 1, . . . , n

Σk=1n Δyrk

UO = dr ; r ∈ Uo}.

Using resource allocation model (9), first

models (2-I) and (2-II) should be solved

to determine MPSS points, for

o = 1, . . . , n. Then, by applying the

obtained optimal solutions to these

models in equation (3) , ρk ; k = 1, . . . , n

are found. On the other hand, for finding

𝜌��;= 1, . . . , n the following models

should be solved: Max η1o − η2o − ζ1o + ζ2o

s. t η1o(yro + Δyro) ≤ ∑ nj=1 λjyrj ; r ∈ DO

ζ1o(xio + Δxio) ≥ ∑ nj=1 λjxij ; i ∈ DDI

η2o(yro + Δyro) ≥ ∑ nj=1 λjyrj ; r ∈ UO

ζ2o(xio + Δxio) ≤ ∑ nj=1 λjxij ; i ∈ DUI

xio ≥ ∑ nj=1 λjxij ; i ∈ NDDI

xio ≤ ∑ nj=1 λjxij ; i ∈ NDUI

∑ nj=1 λj = 1

λj ≥ 0, j = 1, . . . , n

Δ(xik) ∈ ΔDDI

i ∈ DDI, k: 1, . . . , n

Δ(xik) ∈ ΔDDI

i ∈ DUI, k: 1, . . . , n

Δ(yrk) ∈ ΔDO

r ∈ DO, k: 1, . . . , n

Δ(yrk) ∈ ΔUO

i ∈ UO, k: 1, . . . , n

And

Max Σi∈DDI 𝑙𝑖 + Σi∈DUI li + Σr∈DO ℎ𝑟 + Σr∈UO ℎ𝑟 +

Σi∈NDDI li + Σi∈NDUI li s. t η1o

∗(yro + Δyro) = ∑ nj=1 λjyrj − hr ; r ∈ DO

ζ1o∗(xio + Δxio) = ∑ n

j=1 λjxij + li; i ∈ DDI

(10–I)

IJDEA Vol.4, No.2, (2016).737-749


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η2o∗(yro + Δyro) = ∑ n

j=1 λjyrj + ℎ𝑟 ; r ∈ UO

(10 − II)

ζ2o∗(xio + Δxio) = ∑ n

j=1 λjxij − li; i ∈ DUI

xio = ∑ nj=1 λjxij + li; i ∈ NDDI

xio = ∑ nj=1 λjxij − li ; i ∈ NDUI

∑ nj=1 λj = 1

λj ≥ 0; j = 1,… , n

Δ(xik) ∈ ΔDDI ; i ∈ DDI, k: 1,… , n

Δ(xik) ∈ ΔDDI ; i ∈ DUI, k: 1, … , n

Δ(yrk) ∈ ΔDO ; r ∈ DO, k: 1, … , n

Δ(yrk) ∈ ΔUO ; i ∈ UO, k: 1, . . . , n.

These models are the parametric linear

program and their optimal solutions are a

function of ΔxoDDI = (Δxio)i∈DDI,

ΔxoDUI = (Δxio)i∈DUI, Δyo

DO =(Δyro)r∈DO, and Δyo

UO = (Δyro)r∈UO.

Then ρo′ can be obtained as follows:

𝜌�� = Σi∈DDI [ (xio + Δxio)(1 − ζ1o∗ ) +

li∗ ] + Σi∈DUI [ (xio + Δxio)(ζ2o

∗ −

1) + li∗ ] +

Σr∈DO [ (yro + Δyro)(η1o∗ − 1) + hr

∗] +

Σr∈UO [(yro + Δyro)(1 − η2o∗ ) + hr

∗ ] +

Σi∈NDDI li∗ + Σi∈NDUI li

∗. (11)

Theorem 3. Model (2-I) and model(10-I)

are the feasible and bounded problems.

Proof: Refer to Section 5.

Conclusion 1. The dual problems of

models (2-I) and (10-I) have the finite

optimal solutions.

In the following, we introduce the

Theorem 3, which states that the

mathematical relationship between MPSS

points corresponding to DMUo, before

and after resource allocation.

Theorem 4.

If ((λj∗)j=1,...,n; φ1o

∗ ; θ1o∗ ; φ2o

∗ ; θ2o∗ ) is

an optimal solution to the model (2-I),

then for every j = 1, . . . , n, there exists μjo

such that

((λj∗= μjoλj

∗)j=1,...,n; η1o∗ ; ζ1o

∗ ; η2o∗ ; ζ2o

∗ ) is

an optimal solution to the model(10-I);

which is introduced as follows :

λj∗= μjoλj

∗ ; j = 1, . . . , n

η1o∗ = min

r∈DO{ ∑ nj=1 λj

∗yrj

yro + Δyro} ;

ζ1o∗ = max

i∈DDI{ ∑ nj=1 λj

∗xij

xio+Δxio}

η2o∗ = max

r∈UO{ ∑ nj=1 λj

∗yrj

yro + Δyro} ;

ζ2o∗ = min

i∈DUI{ ∑ nj=1λj

∗xij

xio+Δxio}.

Proof: Refer to Section 5.

Conclusion 2. Based on Theorem 1, if

((λj∗)j=1,…,n; φ1o

∗ ; θ1o∗ ; φ2o

∗ ; θ2o∗ ; (tr

∗)r∈DO;

(si∗)i∈DDI; (tr

∗)r∈UO; (si

∗)i∈DUI; (si

∗)i∈NDDI; (si

∗)i∈NDUI) is an optimal solution to a

non-Archimedean form of the model(2-I)

and model (2-II), then:

((λj∗)j=1,...,n; η1o

∗ ; ζ1o∗ ; η2o

∗ ; ζ2o∗ ; (hr

∗)r∈DO;

(li∗)i∈DDI; (hr

∗)r∈UO; (li

∗)i∈DUI; (li

∗)i∈NDDI;

(li∗)i∈NDUI) is an optimal solution to a

non-Archimedean form of the model (10-

I) and model(10-II), such that:

tr∗ = ∑ n

j=1 λj∗yrj − φ1o

∗ yro; r ∈ DO

and tr∗= φ2o

∗ yro − ∑ nj=1 λj

∗yrj; r ∈ UO

si∗ = θ1o

∗ xio − ∑ nj=1 λj

∗xij; i ∈ DDI

and si∗= ∑ n

j=1 λj∗xij − θ2o

∗ xio; i ∈ DUI

si∗ = xio − ∑ n

j=1 λj∗xij; i ∈ NDDIand

hr∗ =

∑ nj=1 λj

∗yrj − minr∈DO

{ ∑ nj=1λj

∗yrj

yro+Δyro}} (yro +

Δyro); r ∈ DO

hr∗= max

r∈UO{ ∑ nj=1λj

∗yrj

yro+Δyro}} (yro + Δyro) −

∑ nj=1 λj

∗yrj; r ∈ UO


1216

li∗ = max

i∈DDI{ ∑ nj=1λj

∗xij

xio+Δxio}} (xio + Δxio) −

∑ nj=1 λj

∗xij; i ∈ DDI

li∗=

∑ nj=1 λj

∗xij − mini∈DUI

{ ∑ nj=1 λj

∗xij

xio+Δxio}} (xio +

Δxio); i ∈ DUI

li∗= xio −∑ n

j=1 λj∗xij; i ∈ NDDI

and li∗= ∑ n

j=1 λj∗xij − xio; i ∈ NDUI.

Based on Theorem 3 and Conclusion 2, ,

there s no need to solve the parametric

models(10-I) and (10-II) and by using the

optimal solution to the models (2-I) and

(2-II), the resource allocation model (9) is

as follows: Max

∑ nk=1 wk (

[Σi∈DDI [xik(1−θ1o

∗ )+si∗ ] +

Σi∈DUI [xik(θ2k∗ −1)+si

∗ ] ]

[Σi∈DDI [ (xik+Δxik)(1−ζ1k

∗ )+li∗ ] +

Σi∈DUI [ (xik+Δxik)(ζ2k∗ −1)+li

∗ ]]

+

[Σr∈DO [yrk(φ1k

∗ − 1) + tr∗ ] +

Σr∈UO [yrk(1 − φ2k∗ ) + tr

∗ ] ]

[Σr∈DO [ (yrk + Δyrk)(η1k

∗ − 1) + hr∗] +

Σr∈UO [ (yrk + Δyrk)(1 − η2k∗ ) + hr

∗ ]]

+ Σi∈NDDI si

∗ + Σi∈NDUI si∗

Σi∈NDDI li∗ + Σi∈NDUI li

∗ )

S.t. (yrk + Δyrk) ≤ ∑ nj=1 λj

kyrj

; r ∈ DO, k = 1, . . . , n

(xik + Δxik) ≥ ∑ nj=1 λj

kxij

; i ∈ DDI, k = 1, . . . , n

(yrk + Δyrk) ≥ ∑ nj=1 λj

kyrj

; r ∈ UO, k = 1, . . . , n

(xik + Δxik) ≤ ∑ nj=1 λj

kxij

; i ∈ DUI, k = 1, . . . , n

xik ≥ ∑ nj=1 λj

kxij

; i ∈ NDDI, k = 1, . . . , n

xik ≤ ∑ nj=1 λj

kxij

; i ∈ NDUI, k = 1, . . . , n

∑ nj=1 λj

k = 1 ; k = 1, . . . , n

Σi∈DDI [xik(1 − θ1o∗ ) + si

∗ ] +

Σi∈DUI [xik(θ2k∗ − 1) + si

∗ ]+

Σr∈DO [yrk(φ1k∗ − 1) + tr

∗ +

Σr∈UO [yrk(1 − φ2k∗ ) + tr

∗ +

Σi∈NDDI si∗ + Σi∈NDUI si

∗

≥

Σi∈DDI [(xik + Δxik)(1 − ζ1k∗ ) + li

∗ ] +

Σi∈DUI [ (xik + Δxik)(ζ2k∗ − 1) + li

∗ ]+

Σr∈DO [ (yrk + Δyrk)(η1k∗ − 1) + hr

∗]+

Σr∈UO [ (yrk + Δyrk)(1 − η2k∗ ) + hr

∗ ]+

+Σi∈NDDI li∗ + Σi∈NDUI li

∗; k = 1, . . . , n (Δx1

DDI, Δx2DDI, . . . , Δxn

DDI) ∈ ΔDDI (Δx1

DUI, Δx2DUI, … , Δxn

DUI) ∈ ΔDUI (Δy1

DO, Δy2DO, . . . , Δyn

DO) ∈ ΔDo

(Δy1UO, Δx2


Remark 2. Note that the objective

function of the above model is fractional

but based on equation (3)

∑ r∈DO [ yrk(φ1k∗ − 1) + tr

∗ ] +∑ r∈UO [yrk(1 − φ2k

∗ ) +

tr∗ ] ∑ i∈NDDI si

∗ + ∑ i∈NDUI si∗ = ρk

∗

therefore the numerator of this function is

a constant value; thus, by considering:

∑ i∈DDI [(xik + Δxik)(1 − ζ1k∗ ) + li

∗ ] +

Σi∈DUI [ (xik + Δxik)(ζ2k∗ − 1) + li

∗ ] + ∑ r∈DO [(yrk + Δyrk)(η1k

∗ − 1) +hr∗] + ∑ r∈UO [ (yrk + Δyrk)(1 −

η2k∗ ) + hr

∗ ] + ∑ i∈NDDI li∗ +

∑ i∈NDUI li∗ = ρ′k ,

we can rewrite this model as follows:

Max ∑ nk=1 wk ρ

′k

S.t. (yrk + Δyrk) ≤ ∑ nj=1 λj

kyrj

; r ∈ DO, k = 1, . . . , n

(xik + Δxik) ≥ ∑ nj=1 λj

kxij

; i ∈ DDI, k = 1, . . . , n


kyrj

; r ∈ UO, k = 1, . . . , n


kyrj

; r ∈ UO, k = 1, . . . , n

xik ≥ ∑ nj=1 λj

kxij

; i ∈ NDDI, k = 1, . . . , n (15)

xik ≤ ∑ nj=1 λj

kxij

(14)

(13)

IJDEA Vol.4, No.2, (2016).737-749


1217

; i ∈ NDUI, k = 1, . . . , n

∑ nj=1 λj

k = 1; k = 1, . . . , n

ρ′k≤ ρk

∗

λjk ≥ 0; k = 1,… , n

(Δx1DDI, Δx2

DDI, . . . , ΔxnDDI) ∈ ΔDDI

(Δx1DUI, Δx2

DUI, … , ΔxnDUI) ∈ ΔDUI

(Δy1DO, Δy2

DO, . . . , ΔynDO) ∈ ΔDo

(Δy1UO, Δx2


Remark 3. For using of formulations

(13), (14), we can use Theorem 4 and

replace the amounts of ζ1k∗ , ζ2k

∗ , η1k∗ , η2k

∗

for k = 1, . . . , n in this model.

However, by considering the sets

ΔDDI, ΔDUI, ΔDO and ΔUO for every

i ∈ DDI, r ∈ DO, we have:

−maxi∈DDI

{∑ nj=1μjλj

∗xij

xio+αio} ≤ −max

i∈DDI{∑ nj=1μjλj

∗xij

xio+Δxio} ≤

−maxi∈DDI

{∑ nj=1μjλj

∗xij

xio+βio}

minr∈DO

{∑ nj=1 μjλj

∗yrj

yro + δro} ≤ min

r∈DO{∑ nj=1 μjλj

∗yrj

yro + Δyro}

≤ minr∈DO

{∑ nj=1 μjλj

∗yrj

yro + γro}

and for every i ∈ DUI, i ∈ UO, we have:

mini∈DUI

{∑ nj=1 μjλj

∗xij

xio + β′io

} ≤ mini∈DUI

{∑ nj=1 μjλj

∗xij

xio + Δxio}

≤ mini∈DUI

{∑ nj=1 μjλj

∗xij

xio + α′io

}

−maxr∈UO

{∑ nj=1 μjλj

∗yrj

yro + γ′ro

} ≤

−maxr∈UO

{∑ nj=1 μjλj

∗yrj

yro + Δyro}

≤ −maxr∈UO

{∑ nj=1 μjλj

∗yrj

yro + δ′ro}.

Then, for achieving the minimum value's

objective function of model (12), we

replace (−ζ1o∗ ) with

(−maxi∈DDI{∑ nj=1μjλj

∗xij

xio+αio}) and (η1o

∗ ) with

(minr∈DO{∑ nj=1μjλj

∗yrj

yro+δro}) and (ζ2o

∗ ) with

(mini∈DUI{∑ nj=1μjλj

∗xij

xio+β′io}) and (−η2o

∗ ) with

(−maxr∈UO{∑ nj=1μjλj

∗yrj

yro+γ′ro}).

4. Application Examples: In this section the proposed method in

Section 3 is used for allocating resources

in two real world situations. In one of

these examples, there are data of desirable

and undesirable type and in the other

example in addition to these types, some

of the data are of discretionary and non

discretionary type.

Example 1: Here, we illustrate the resource allocation

method discussed in this paper through

the analysis data from petroleum

companies. The data set consists of 20 gas

companies located in 18 regions in Iran.

The data for this analysis are derived

from operations during 2005. There are

six variables from the data set as inputs

and outputs in this example. Inputs

include capital (x1), number of staff (x2),

and operational costs (excluding staff

costs) (x3) and outputs include number of

subscribers (y1), length of gas network

(y2) and the sold-out gas income (y3).

Table 1 contains a listing of the original

data.


1218

Table1:Dat a of Inputs and outputs for Iranian gascompanies

There is central company (DM) which

supervises the above gas companies. This

company has a notable additional capital

and staff. The amount of additional

capital is 15000 and the number of

additional staff is 300. The central

company wants to allocate them to those

companies so that the number of

subscribers reaches 500684 and the

amounts of sold-out gas income reaches

2936415. The proposed method is

applied in order to determine the

contribution of each of company.

Here one must note that some data are

desirable and others are undesirable. For

example capital and operational costs are

discretionary desirable input and non

discretionary desirable input,respectively.

Also, the number of staff is undesirable

Input because one of the DM's aims is to

help solve the unemployment issue and

the number of subscribers and sold-out

gas income are desirable outputs but the

length of gas network is undesirable

output.

First the model (2 − I) is solved and the

following results are achieved: (Table2)

DMU 𝑥1 𝑥2 𝑥3 𝑦1 𝑦2 𝑦3 1 124313 129 198598 30242 565 61836

2 67545 117 131649 14139 153 46233

3 47208 165 228730 13505 211 42094

4 43494 106 165470 8508 114 44195

5 48308 141 180866 7478 248 45841

6 55959 146 194470 10818 230 136513

7 40605 145 179650 6422 127 70380

8 61402 87 94226 18260 182 36592

9 87950 104 91461 22900 170 47650

10 33707 114 88640 3326 85 13410

11 100304 254 292995 14780 318 79833

12 94286 105 98302 19105 273 32553

13 67322 224 287042 15332 241 172316

14 102045 104 155514 18082 441 30004

15 177430 401 528325 77564 801 201529

16 221338 1094 1186905 44136 803 840446

17 267806 1079 1323325 27690 251 832616

18 160912 444 648685 45882 816 251770

19 177214 801 909539 72676 654 34158

20 146325 686 545115 19839 177 341585

IJDEA Vol.4, No.2, (2016).737-749


1219

Table2: The results of model (2 − I)

Now, by applying Theorem 4 we obtain

μjo∗ and then the λj

∗ are achieved as

follows:

λj∗= {

0.99 ; j = 100 ; j = 10 For o = 1;

λj∗= {

0.97 ; j = 100 ; j = 10 For o = 2

λj∗= {

1.004 ; j = 100 ; j = 10 For o = 3;

λj∗= {

0.63 ; j = 90.37 ; j = 100 ; j = 10,9

For o = 4

λj∗= {

0.562 ; j = 90.454 ; j = 100 ; j = 10,9

For o = 5;

λj∗= {

0.99 ; j = 100 ; j = 10 For o = 6

λj∗= {

0.043 ; j = 90.973 ; j = 100 ; j = 10,9

For o = 7;

λj∗= {

0.95 ; j = 100 ; j = 10 For o = 8

λj∗= {

0.98 ; j = 100 ; j = 10 For o = 9;

λj∗= {

1 ; j = 100 ; j = 10 For o = 10

λj∗= {

0.99 ; j = 90 ; j = 9 For o = 11;

λj∗= {

0.99 ; j = 90 ; j = 9 For o = 12

λj∗= {

0.99 ; j = 100 ; j = 10 For o = 13;

λj∗= {

1 ; j = 90 ; j = 9 For o = 14

DMU 𝜑1𝑘 𝜃1𝑘 𝜑2𝑘 𝜃2𝑘 1 0.24151 0.48936 0.18967 1.95175 2 0.34528 0.65612 0.61221 1.4349 3 1.1226 1.2147 0.86304 1.4014 4 1.9473 1.8704 1.2782 1.9153 5 2.0524 1.6632 0.56955 1.5853

6 0.65558 1.0689 0.46230 1.6891 7 1.1361 1.4290 0.83687 1.5695

8 0.19321 0.57140 0.50334 1.1301 9 0.14970 0.39117 0.50334 1.1301

10 1 1 1 1 11 1.8896 1.1112 0.71563 1.2670 12 1.1962 0.94214 0.62310 1.0741

13 0.82394 1.1767 0.6085 1.5369 14 1.2426 0.94265 0.38773 1.79014 15 0.24795 0.80135 0.21675 1.6583 16 0.83067 1.0842 0.41239 0.91003 17 1 1 1 1 18 0.75967 0.95507 0.38304 1.6187 19 0.27298 0.84967 0.27064 0.85643

20 1 1 1.475 1


1220

λj∗= {

1 ; j = 100 ; j = 10 For o = 15;

λj∗= {

1.02 ; j = 190.511 ; j = 200 ; j = 19,20

Foro = 16

λj∗= {

1 ; j = 170 ; j = 17 For o = 17;

λj∗= {

1 ; j = 200 ; j = 20 For o = 18

λj∗= {

1.18 ; j = 200 ; j = 20 For o = 19;

λj∗= {

1 ; j = 200 ; j = 20 For o = 20

and we have: (Table3) and according to Conclusion 2 and

Remark 3, we obtain h1∗ , h2

∗, h3

∗ and l1∗ ,

l2∗ ,l3∗ . Then we replace these parameters in

the model (15), the following results are

achieved: (Table4)

Table3: Efficien cyscores for DMUs afterre source allocation DMU

DMU 𝜂1𝑘 𝜂2𝑘 𝜁1𝑘 𝜁2𝑘 1 0.54 0.16 0.27 0.25 2 0.61 0.75 0.49 0.25 3 0.064 0.51 0.72 0.23 4 0.165 1.94 1.57 0.25 5 0.154 0.65 1.34 0.24 6 0.044 0.45 0.6 0.24 7 0.014 1.07 0.91 0.24 8 0.062 0.58 0.52 0.26 9 0.060 0.65 0.38 0.26

10 0.074 2 1.005 0.26 11 0.192 0.62 0.88 0.18 12 0.235 0.73 0.93 0.24 13 0.039 0.43 0.5 0.21 14 0.24 0.43 0.86 0.24 15 0.031 0.11 0.19 0.14 16 0.52 1 1.16 0.82 17 0.47 1.2 1 0.77 18 0.26 0.23 0.91 0.89 19 0.23 0.34 0.98 0.83 20 0.39 1.32 1 0.8

IJDEA Vol.4, No.2, (2016).737-749


1221

Table4:Data of additional inputs and additional outputs for DMUs DMU

According to the optimal resource

allocation plan, company 17 and company

10 will receive less resources than the

other companies because these units are

more efficient than the other DMUs.

Example 2: A manufacturer of

agricultural products manages three

farms. These farms consume resources

such as land, seeds, animal manure,

carbon monoxide (co), which is a toxic

gases and etc to produce several types of

fruits and vegetables. The information of

these farms, such as the quantities of

consumed resources and produced

products, is introduced in Table 5.

Table5: Consumed inputs and produced outputs of the farms farm1

DMU Δx1k Δx2k Δy1k Δy2k Δy3k 1 402.8 54.36722 0 0 26726.38

2 887.15 25.16787 2985.19 7.303861 1552.059

3 887.15 5.797695 144.9329 0 2391.259

4 887.15 60.58406 0 15.55526 0

5 887.15 2.394052 1717.488 0 18906.59

6 887.15 0 0 0 0

7 887.15 1.011556 0 109654.4 0

8 887.15 0.6775440 152.3884 0 423.7117

9 887.15 0 0 0 0

10 0 0 0 0 0

11 887.15 0 0 84248.1 0

12 887.15 0 0 486754.3 0

13 0 0 0 0 0

14 887.15 0 0 1243889 0

15 887.15 150 5000 0 50000

16 887.15 0 0 819897.6 0

17 402.8 0 0 0 0

18 887.15 0 0 1047185 0

19 887.15 0 0 6461624 0

20 887.15 0 0 0 0

farm1 farm2 farm3

land(m2) 100 150 200

Seeds (kg) 12 17 20

rainfall(mm) 30 20 18

Car bonmonoxide(m3) 1 1.5 2

animalmanure(kg) 5 8 12

desirablefruits(kg) 20 28 32

undesirablefruits(kg) 3 4 5

Desirablevegetables(kg) 15 20 28

undesirablefruits(kg) 2 3 2


1222

This manufacturer has a significant

manure and seeds excess. The DM wants

to allocate them to the farms so that the

production level reaches a special level.

This paper's method is applied to

determine the allocation resources to each

farm.

For this aim, we consider the farms as

DMUs; the land, seeds, animal manure

and carbon monoxide (co) are considered

as inputs and the produced products are

assumed as outputs. Note that some data

are desirable and others are undesirable.

Also two classes for inputs can be

considered: discretionary and non

discretionary inputs. The decomposition

of data is as follows: (Table6) The manufacturer expresses the following

conditions for resource allocation:

𝛥𝐷𝐷𝐼 = {(𝛥𝑥𝑖1, 𝛥𝑥𝑖2, 𝛥𝑥𝑖3);

−0.4𝑥𝑖𝑘 ≤ 𝛥𝑥𝑖𝑘 ≤ 0.5𝑥𝑖𝑘; 𝑘 = 1,2,3; 𝑖 = 1,2; ∑ 3𝑘=1 𝛥𝑥1𝑘 = 120 ∑ 3

𝑘=1 𝛥𝑥2𝑘 = 10} 𝛥𝐷𝑈𝐼 = {(𝛥𝑥51, 𝛥𝑥52, 𝛥𝑥53); −0.2𝑥5𝑘 ≤ 𝛥𝑥5𝑘 ≤ 0.3𝑥5𝑘; 𝑘 = 1,2,3; ∑ 3𝑘=1 𝛥𝑥5𝑘 = 2}

𝛥𝐷𝑂 = {(𝛥𝑦𝑟1, 𝛥𝑦𝑟2, 𝛥𝑦𝑟3); −0.2𝑦𝑟𝑘 ≤ 𝛥𝑦𝑟𝑘 ≤ 0.3𝑦𝑟𝑘; 𝑘 = 1,2,3; 𝑟 = 1,3; ∑ 3𝑘=1 𝛥𝑦1𝑘 = 8 ∑ 3

𝑘=1 𝛥𝑦3𝑘 = 5} 𝛥𝑈𝑂 = {(𝛥𝑦𝑟1, 𝛥𝑦𝑟2, 𝛥𝑦𝑟3); −0.1𝑦𝑟𝑘 ≤ 𝛥𝑦𝑟𝑘 ≤ 0.3y𝑟𝑘; 𝑘 = 1,2,3; 𝑟 = 2,4; ∑ 3𝑘=1 𝛥𝑦2𝑘 = 2 ∑ 3

𝑘=1 𝛥𝑦4𝑘 = 2}.

After applying the algorithm which is

proposed in this paper, the following

results are achieved. (Table7)

Table6: The decomposed data 𝐃𝐢𝐬𝐜𝐫𝐞𝐭𝐢𝐨𝐧𝐚𝐫𝐲𝐃𝐞𝐬𝐢𝐫𝐚𝐛𝐥𝐞 land and

𝐈𝐧𝐩𝐮𝐭𝐬(𝐃𝐃𝐈) seeds

𝐍𝐨𝐧𝐝𝐢𝐬𝐜𝐫𝐞𝐭𝐢𝐨𝐧𝐚𝐫𝐲 𝐃𝐞𝐬𝐢𝐫𝐚𝐛𝐥𝐞

𝐈𝐧𝐩𝐮𝐭(𝐍𝐃𝐈) rainfall

𝐃𝐢𝐬𝐜𝐫𝐞𝐭𝐢𝐨𝐧𝐚𝐫𝐲 𝐔𝐧𝐝𝐞𝐬𝐢𝐫𝐚𝐛𝐥𝐞

𝐈𝐧𝐩𝐮𝐭(𝐃𝐔𝐈) 𝐚𝐧𝐢𝐦𝐚𝐥 manure

𝐍𝐨𝐧𝐝𝐢𝐬𝐜𝐫𝐞𝐭𝐢𝐨𝐧𝐚𝐫𝐲𝐔𝐧𝐝𝐞𝐬𝐢𝐫𝐚𝐛𝐥𝐞

𝐈𝐧𝐩𝐮𝐭(𝐍𝐔𝐈) co

𝐔𝐧𝐝𝐞𝐬𝐢𝐫𝐚𝐛𝐥𝐞𝐎𝐮𝐭𝐩𝐮𝐭(𝐃𝐎) 𝐔𝐧𝐝𝐞𝐬𝐢𝐫𝐚𝐛𝐥𝐞 𝐟𝐫𝐮𝐢𝐭𝐬 𝐚𝐧𝐝 𝐯𝐞𝐠𝐞𝐭𝐚𝐛𝐥𝐞𝐬

Table7: The result of the propose

farm1 farm2 farm3

ΔDDIland(m2) 120 0 0

ΔDDISeeds (kg) 9 1 0

ΔDUIanimalmanure(kg) 0 2 0

ΔDO desirablefruits(kg) 2 6 0

ΔUOundesirablefruits(kg) 2 0 0

ΔDOdesirablevegetables(kg) 0.5 4.5 0

ΔUOundesirablevegetables(kg) 1.5 0.5 0

IJDEA Vol.4, No.2, (2016).737-749


1223

5. Appendix

In this section Theorem 3 and Theorem 4

which were introduced in Section 3 are

proved.

Proof of Theorem 3: Suppose that:

𝜆𝑗 = {0 ; 𝑗 ≠ 𝑜1 ; 𝑗 = 𝑜

𝜂1𝑜 = 𝜁2𝑜 = 0

and

𝜁1𝑜 = 𝑚𝑎𝑥𝑖∈𝐷𝐷𝐼

{𝑥𝑖𝑜

𝑥𝑖𝑜 + 𝛥𝑥𝑖𝑜} ,

𝜂2𝑜 = 𝑚𝑎𝑥𝑟∈𝑈𝑂

{𝑦𝑟𝑜

𝑦𝑟𝑜 + 𝛥𝑦𝑟𝑜}.

This is a feasible solution to the

model(10-I). On the other hand, the

constraints of model(10-I) imply that

there is an upper bound for the amount of

objective function in every feasible

solutions of model (10-I), as follows:

𝜂1𝑜 − 𝜁1𝑜 − 𝜂2𝑜 +

𝜁2𝑜 ≤ 𝑚𝑖𝑛𝑟∈𝐷𝑂

{∑ 𝑛𝑗=1 𝜆𝑗𝑦𝑟𝑗

𝑦𝑟𝑜 + 𝛥𝑦𝑟𝑜}

−𝑚𝑎𝑥𝑖∈𝐷𝐷𝐼

{∑ 𝑛𝑗=1 𝜆𝑗𝑥𝑖𝑗

𝑥𝑖𝑜 + 𝛥𝑥𝑖𝑜}

−𝑚𝑎𝑥𝑟∈𝑈𝑂

{∑ 𝑛𝑗=1 𝜆𝑗𝑦𝑟𝑗

𝑦𝑟𝑜 + 𝛥𝑦𝑟𝑜}

+𝑚𝑖𝑛𝑖∈𝐷𝑈𝐼

{∑ 𝑛𝑗=1 𝜆𝑗𝑥𝑖𝑗

𝑥𝑖𝑜 + 𝛥𝑥𝑖𝑜}.

Therefore model (10-I) is a bounded

problem. Similarly, it can be shown that

model (2-I) is also feasible and bounded.

Proof of Theorem 4. Suppose that set

𝐽 = {1, . . . . , 𝑛} is decomposed as follows: 𝐽 = 𝐽1⋃ 𝐽2 such that:

𝐽1 = {𝑗 ∈ 𝐽| 𝑚𝑎𝑥𝑖∈𝑁𝐷𝑈𝐼

{𝑥𝑖𝑜𝑥𝑖𝑗} ≤ 𝑚𝑖𝑛

𝑖∈𝑁𝐷𝐷𝐼{𝑥𝑖𝑜𝑥𝑖𝑗}}.

Note that if 𝑁𝐷𝑈𝐼 = ∅ then 𝐽1 = 𝐽 and

𝐽2 = 𝐽.

and

𝐽2 = {𝑗 ∈ 𝐽| 𝑚𝑎𝑥𝑖∈𝑁𝐷𝑈𝐼

{𝑥𝑖𝑜𝑥𝑖𝑗} > 𝑚𝑖𝑛

𝑖∈𝑁𝐷𝐷𝐼{𝑥𝑖𝑜𝑥𝑖𝑗}}.

Set 𝐽3 = {𝑗 ∈ 𝐽1|𝑚𝑎𝑥𝑖∈𝑁𝐷𝑈𝐼{𝑥𝑖𝑜

𝑥𝑖𝑗} +

𝑚𝑖𝑛𝑖∈𝑁𝐷𝐷𝐼{𝑥𝑖𝑜

𝑥𝑖𝑗} > 1}. 𝐽3 ≠ ∅, Because

𝑜 ∈ 𝐽3, for 𝑗 = 1, . . . , 𝑛, 𝜇𝑗 can be defined

as follows: 𝜇𝑗𝑜

=

{

𝑚𝑎𝑥𝑖∈𝑁𝐷𝑈𝐼

{𝑥𝑖𝑜

𝑥𝑖𝑗} + 𝑚𝑖𝑛

𝑖∈𝑁𝐷𝐷𝐼{𝑥𝑖𝑜

𝑥𝑖𝑗}

∑ 𝑗∈𝐽3 ( 𝑚𝑎𝑥𝑖∈𝑁𝐷𝑈𝐼{𝑥𝑖𝑜



𝑥𝑖𝑗})𝜆𝑗

∗; 𝑗 ∈ 𝐽3

0; 𝑗 ∈ 𝐽 − 𝐽3

Here, we will show that ((𝜆��∗=

𝜇𝑗𝑜𝜆𝑗∗)𝑗=1,...,𝑛; 𝜂1𝑜

∗ ; 𝜁1𝑜∗ ; 𝜂2𝑜

∗ ; 𝜁2𝑜∗ ) is a

feasible solution to model(10-I). For this

aim, we are going to investigate the

constraints of model(10-I).

We note that for 𝑗 ∈ 𝐽3,

𝜇𝑗𝑜 =𝑚𝑎𝑥𝑖∈𝑁𝐷𝑈𝐼

{𝑥𝑖𝑜



𝑥𝑖𝑗}

∑ 𝑗∈𝐽3 ( 𝑚𝑎𝑥𝑖∈𝑁𝐷𝑈𝐼{𝑥𝑖𝑜




∗

=


{𝑥𝑖𝑜𝑥𝑖𝑗

}+ 𝑚𝑖𝑛𝑖∈𝑁𝐷𝐷𝐼


}

∑ 𝑗∈𝐽3 𝜆𝑗∗

∑ 𝑗∈𝐽3 ( 𝑚𝑎𝑥𝑖∈𝑁𝐷𝑈𝐼{𝑥𝑖𝑜𝑥𝑖𝑗

}+ 𝑚𝑖𝑛𝑖∈𝑁𝐷𝐷𝐼


})𝜆𝑗∗

∑ 𝑗∈𝐽3 𝜆𝑗∗

.

Based on the definition of 𝐽3 and that

((𝜆𝑗∗)𝑗=1,...,𝑛; 𝜑1𝑜

∗ ; 𝜃1𝑜∗ ; 𝜑2𝑜

∗ ; 𝜃2𝑜∗ ) being

an optimal solution to the model (2-I), we

know that:

∑ 𝑗∈𝐽3( 𝑚𝑎𝑥𝑖∈𝑁𝐷𝑈𝐼

{𝑥𝑖𝑜




∗

∑ 𝑗∈𝐽3 𝜆𝑗∗ > 1

and this implies that for every 𝑗 ∈ 𝐽3,

𝑚𝑎𝑥𝑖∈𝑁𝐷𝑈𝐼{𝑥𝑖𝑜𝑥𝑖𝑗}

∑ 𝑗∈𝐽3 𝜆𝑗∗ ≤ 𝜇𝑗𝑜 ≤

𝑚𝑖𝑛𝑖∈𝑁𝐷𝐷𝐼{𝑥𝑖𝑜𝑥𝑖𝑗}

∑ 𝑗∈𝐽3 𝜆𝑗∗ .

For every 𝑖 ∈ 𝑁𝐷𝐷𝐼, we have:


1224

∑ 𝑛𝑗=1 𝜆��

∗𝑥𝑖𝑗 = ∑ 𝑛

𝑗=1 𝜇𝑗𝑜𝜆𝑗∗𝑥𝑖𝑗 =

∑ 𝑗∈𝐽3 𝜇𝑗𝑜𝜆𝑗∗𝑥𝑖𝑗 ≤ ∑ 𝑗∈𝐽3

𝑚𝑖𝑛𝑖∈𝑁𝐷𝐷𝐼

{𝑥𝑖𝑜𝑥𝑖𝑗}

∑ 𝑗∈𝐽3 𝜆𝑗∗ 𝜆𝑗

∗𝑥𝑖𝑗

≤1

∑ 𝑗∈𝐽3 𝜆𝑗∗∑

𝑗∈𝐽3

𝑥𝑖𝑜𝑥𝑖𝑗

𝜆𝑗∗𝑥𝑖𝑗 ≤

𝑥𝑖𝑜∑ 𝑗∈𝐽3 𝜆𝑗

∗∑

𝑗∈𝐽3

𝜆𝑗∗

= 𝑥𝑖𝑜 .

Similarly, for every 𝑖 ∈ 𝑁𝐷𝑈𝐼, we have:

∑

𝑛

𝑗=1

𝜆��∗𝑥𝑖𝑗 =∑

𝑛

𝑗=1

𝜇𝑗𝑜𝜆𝑗∗𝑥𝑖𝑗 = ∑

𝑗∈𝐽3

𝜇𝑗𝑜𝜆𝑗∗𝑥𝑖𝑗

≥ ∑

𝑗∈𝐽3


{𝑥𝑖𝑜

𝑥𝑖𝑗}

∑ 𝑗∈𝐽3 𝜆𝑗∗ 𝜆𝑗

∗𝑥𝑖𝑗

≥1

∑ 𝑗∈𝐽3 𝜆𝑗∗∑

𝑗∈𝐽3

𝑥𝑖𝑜𝑥𝑖𝑗

𝜆𝑗∗𝑥𝑖𝑗 ≥

𝑥𝑖𝑜∑ 𝑗∈𝐽3 𝜆𝑗

∗∑

𝑗∈𝐽3

𝜆𝑗∗

= 𝑥𝑖𝑜 .

For every 𝑟 ∈ 𝐷𝑂:

𝜂1𝑜∗ (𝑦𝑟𝑜 + 𝛥𝑦𝑟𝑜) = 𝑚𝑖𝑛

𝑟∈𝐷𝑂{ ∑ 𝑛𝑗=1 𝜆��

∗𝑦𝑟𝑗

𝑦𝑟𝑜 + 𝛥𝑦𝑟𝑜}(𝑦𝑟𝑜

+ 𝛥𝑦𝑟𝑜)

≤∑ 𝑛𝑗=1 𝜆��

∗𝑦𝑟𝑗

𝑦𝑟𝑜 + Δ𝑦𝑟𝑜(𝑦𝑟𝑜 + 𝛥𝑦𝑟𝑜)

=∑

𝑛

𝑗=1

𝜆��∗𝑦𝑟𝑗

and for every 𝑖 ∈ 𝐷𝐷𝐼:

𝜁1𝑜∗ (𝑥𝑖𝑜 + 𝛥𝑥𝑖𝑜) = 𝑚𝑎𝑥

𝑖∈𝐷𝐷𝐼{ ∑ 𝑛𝑗=1 𝜆��

∗𝑥𝑖𝑗

𝑥𝑖𝑜 + 𝛥𝑥𝑖𝑜}(𝑥𝑖𝑜

+ 𝛥𝑥𝑖𝑜)

≥∑ 𝑛𝑗=1 𝜆��

∗𝑥𝑖𝑗

𝑥𝑖𝑜 + 𝛥𝑥𝑖𝑜(𝑥𝑖𝑜 + 𝛥𝑥𝑖𝑜)

=∑

𝑛

𝑗=1

𝜆��∗𝑥𝑖𝑗 .

For 𝑟 ∈ 𝑈𝑂:

𝜂2𝑜∗ (𝑦𝑟𝑜 + 𝛥𝑦𝑟𝑜) = 𝑚𝑎𝑥

𝑟∈𝑈𝑂{ ∑ 𝑛𝑗=1 𝜆��

∗𝑦𝑟𝑗

𝑦𝑟𝑜 + 𝛥𝑦𝑟𝑜}(𝑦𝑟𝑜

+ 𝛥𝑦𝑟𝑜)

≥∑ 𝑛𝑗=1 𝜆��

∗𝑦𝑟𝑗

𝑦𝑟𝑜 + 𝛥𝑦𝑟𝑜(𝑦𝑟𝑜 + 𝛥𝑦𝑟𝑜)

=∑

𝑛

𝑗=1

𝜆��∗𝑦𝑟𝑗

and for every 𝑖 ∈ 𝐷𝑈𝐼:

𝜁2𝑜∗ (𝑥𝑖𝑜 + 𝛥𝑥𝑖𝑜) = 𝑚𝑖𝑛

𝑖∈𝐷𝑈𝐼{ ∑ 𝑛𝑗=1 𝜆��

∗𝑥𝑖𝑗

𝑥𝑖𝑜 + 𝛥𝑥𝑖𝑜}(𝑥𝑖𝑜

+ 𝛥𝑥𝑖𝑜)

≤∑ 𝑛𝑗=1 𝜆��

∗𝑥𝑖𝑗

𝑥𝑖𝑜 + 𝛥𝑥𝑖𝑜}(𝑥𝑖𝑜 + 𝛥𝑥𝑖𝑜)

≤ ∑

𝑛

𝑗=1

𝜆��∗𝑥𝑖𝑗

and it is clear that:

𝜇𝑗𝑜 ≥ 0; 𝑗 = {1,2, . . . , 𝑛} 𝑎𝑛𝑑 ∑

𝑛

𝑗=1

𝜆𝑗∗

= ∑

𝑗∈𝐽3

𝜆𝑗∗ = 1.

Then 𝜆��∗= 𝜇𝑗𝑜𝜆𝑗

∗)𝑗=1,...,𝑛; 𝜂1𝑜∗ ; 𝜁1𝑜

∗ ; 𝜂2𝑜∗ ; 𝜁2𝑜

∗ )

is a feasible solution to the model (10-I).

Now, suppose that

((𝜆��𝑗=1,...,𝑛∗∗

; 𝜂1𝑜∗∗ ; 𝜁1𝑜

∗∗; 𝜂2𝑜∗∗ ; 𝜁2𝑜

∗∗) is an

optimal solution to the model(10-I), then

we have:

𝜂1𝑜∗∗ ≥ 𝜂1𝑜

∗ , 𝜁1o∗∗ ≤ 𝜁1𝑜

∗ , 𝜂2𝑜∗∗ ≤ 𝜂2𝑜

∗ , ; 𝜁2𝑜∗∗ ≥ 𝜁2𝑜

∗ . In the following, we will show that:

𝜂1𝑜∗∗ ≤ 𝜂1𝑜

∗ , 𝜁1𝑜∗∗ ≥ 𝜁1𝑜

∗ , 𝜂2𝑜

∗∗ ≥ 𝜂2𝑜∗ , ; 𝜁2𝑜

∗∗ ≤ 𝜁2𝑜∗

which implies that: 𝜂1𝑜∗∗ = 𝜂1𝑜

∗ , 𝜁1𝑜∗∗ = 𝜁1𝑜

∗ , 𝜂2𝑜∗∗ = 𝜂2𝑜

∗ , 𝜁2𝑜∗∗ = 𝜁2𝑜

∗ .

or in other words ((𝜆��∗= 𝜇𝑗𝑜𝜆𝑗

∗)𝑗=1,…,𝑛; 𝜂1𝑜∗ ;

𝜁1𝑜∗ ; 𝜂2𝑜

∗ ; 𝜁2𝑜∗ ) is an optimal solution to

the model(2-I). For this aim, we first

prove that

2 −𝑚𝑎𝑥𝑟∈𝐷𝑂

{𝑦𝑟𝑜

∑ 𝑛𝑗=1 𝜆𝑗

∗𝑦𝑟𝑗}𝜑1𝑜

∗ = 1 𝑎𝑛𝑑

2 − 𝑚𝑖𝑛𝑟∈𝑈𝑂

{𝑦𝑟𝑜



∗ = 1

2 − 𝑚𝑖𝑛𝑖∈𝐷𝐷𝐼

{𝑥𝑖𝑜


∗𝑥𝑖𝑗} 𝜃1𝑜

∗ = 1 𝑎𝑛𝑑

2 − 𝑚𝑎𝑥𝑖∈𝐷𝑈𝐼

{𝑥𝑖𝑜


∗𝑥𝑖𝑗}𝜃2𝑜

∗ = 1

because

((𝜆𝑗∗)𝑗=1,...,𝑛; 𝜑1𝑜

∗ ; 𝜃1𝑜∗ ; 𝜑2𝑜

∗ ; 𝜃2𝑜∗ ) is an

optimal solution to the model(2-I), then

we have:

𝜑1𝑜∗ 𝑦𝑟𝑜 ≤∑

𝑛

𝑗=1

𝜆𝑗∗𝑦𝑟𝑗 ; 𝑟 ∈ 𝐷𝑂

IJDEA Vol.4, No.2, (2016).737-749


1225

𝜃1𝑜∗ 𝑥𝑖𝑜 ≥∑

𝑛

𝑗=1

𝜆𝑗∗𝑥𝑖𝑗 ; 𝑖 ∈ 𝐷𝐷𝐼

𝜑2𝑜∗ 𝑦𝑟𝑜 ≥∑

𝑛

𝑗=1

𝜆𝑗∗𝑦𝑟𝑗 ; 𝑟 ∈ 𝑈𝑂

𝜃2𝑜∗ 𝑥𝑖𝑜 ≤∑

𝑛

𝑗=1

𝜆𝑗∗𝑥𝑖𝑗 𝑖 ∈ 𝐷𝑈𝐼

But by using the complementary theorem,

there exists 𝑘1 ∈ 𝐷𝑂, 𝑘2 ∈ 𝐷𝐷𝐼, 𝑘3 ∈ 𝑈𝑂

and 𝑘4 ∈ 𝐷𝑈𝐼 such that:

𝜑1𝑜∗ 𝑦 𝑘1𝑜 =∑

𝑛

𝑗=1

𝜆𝑗∗𝑦 𝑘1𝑗

𝜃1𝑜∗ 𝑥 𝑘2𝑜 =∑

𝑛

𝑗=1

𝜆𝑗∗𝑥 𝑘2𝑗

𝜑2𝑜∗ 𝑦 𝑘3𝑜 =∑

𝑛

𝑗=1

𝜆𝑗∗𝑦 𝑘3𝑗

𝜃2𝑜∗ 𝑥 𝑘4𝑜 =∑

𝑛

𝑗=1

𝜆𝑗∗𝑥 𝑘4𝑗

because if, for example, there is no

binding constraint of 𝑟 ∈ 𝐷𝑂 in the

optimal solution, then their corresponding

dual variables(𝛾𝑟; 𝑟 ∈ 𝐷𝑂) will be zero

and the constraint corresponding 𝜑1𝑜 in

the dual problem to the model(2-I) will be

as follows:

∑

𝑟∈𝐷𝑂

(𝛾𝑟𝑦𝑟𝑜) = 1

or in other words 0 = 1 and this is a

contradiction. Similarly, we can show that

there exist 𝑘2 ∈ 𝐷𝐷𝐼, 𝑘3 ∈ 𝑈𝑂 and

𝑘4 ∈ 𝐷𝑈𝐼. However, based on the above

explanations , it is obvious that: 𝑦 𝑘1𝑜


∗𝑦 𝑘1𝑗𝜑1𝑜∗ = 𝑚𝑎𝑥

𝑟∈𝐷𝑂

𝑦𝑟𝑜∑ n𝑗=1 𝜆𝑗

∗𝑦𝑟𝑗𝜑1𝑜∗ ,

𝑥 𝑘2𝑜


∗𝑥 𝑘2𝑗𝜃1𝑜∗ = 𝑚𝑖𝑛

𝑖∈𝐷𝐷𝐼

𝑥𝑖𝑜∑ 𝑛𝑗=1 𝜆𝑗

∗𝑥𝑖𝑗𝜃1𝑜∗

𝑦 𝑘3𝑜∑ 𝑛𝑗=1 𝜆𝑗

∗𝑦 𝑘3𝑗𝜑2𝑜∗ = 𝑚𝑖𝑛

𝑟∈𝑈𝑂

𝑦𝑟𝑜∑ 𝑛𝑗=1 𝜆𝑗

∗𝑦𝑟𝑗𝜑2𝑜∗ ,

𝑥 𝑘4𝑜∑ 𝑛𝑗=1 𝜆𝑗

∗𝑥 𝑘4𝑗𝜃2𝑜∗ = 𝑚𝑎𝑥

𝑖∈𝐷𝑈𝐼

𝑥𝑖𝑜∑ 𝑛𝑗=1 𝜆𝑗

∗x𝑖𝑗𝜃2𝑜∗

thus:

2 −𝑚𝑎𝑥𝑟∈𝐷𝑂

{𝑦𝑟𝑜



∗ = 1 𝑎𝑛𝑑


{𝑦𝑟𝑜



∗ = 1


{𝑥𝑖𝑜


∗𝑥𝑖𝑗} 𝜃1𝑜

∗ = 1 𝑎𝑛𝑑


{𝑥𝑖𝑜



∗ = 1

then, we can rewrite:

𝜂1𝑜∗ =

𝑚𝑖𝑛𝑟∈𝐷O

{ ∑ 𝑛𝑗=1𝜇𝑗𝑜𝜆𝑗

∗𝑦𝑟𝑗

𝑦𝑟𝑜+𝛥𝑦𝑟𝑜}

2 − 𝑚𝑎𝑥𝑟∈𝐷𝑂

{𝑦𝑟𝑜



∗

𝜂2𝑜∗ =

𝑚𝑎𝑥𝑟∈𝑈𝑂


∗𝑦𝑟𝑗



{𝑦𝑟𝑜



∗

𝜁2𝑜∗ =

𝑚𝑖𝑛𝑖∈𝐷𝑈𝐼


∗𝑥𝑖𝑗

𝑥𝑖𝑜+𝛥𝑥𝑖𝑜}


{𝑥𝑖𝑜



∗

𝜁1𝑜∗ =

𝑚𝑎𝑥𝑖∈𝐷𝐷𝐼


∗𝑥𝑖𝑗



{𝑥𝑖𝑜



∗.

Here, we will show that

((𝜆𝑗∗)𝑗=1,...,𝑛, 𝜑1��, 𝜃1��, 𝜑2��, 𝜃2��) such that:

𝜑1�� =2 −

𝑚𝑖𝑛𝑟∈𝐷𝑂

{ ∑ 𝑛𝑗=1𝜆��

∗∗y𝑟𝑗


𝜂10∗∗

𝑚𝑎𝑥𝑟∈𝐷𝑂

{𝑦𝑟𝑜


∗𝑦𝑟𝑗}

𝜃1�� =2 −


{∑ 𝑛𝑗=1𝜆��

∗∗𝑥𝑖𝑗


𝜁1𝑜∗∗

𝑚𝑖𝑛𝑖∈𝐷𝐷𝐼

{𝑥𝑖𝑜

∑ 𝑛𝑗=1𝜆𝑗

∗𝑥𝑖𝑗}

𝜑2�� =2 −

𝑚𝑎𝑥𝑟∈𝑈𝑂

{ ∑ 𝑛𝑗=1𝜆��

∗∗𝑦𝑟𝑗


𝜂20∗∗

𝑚𝑖𝑛𝑟∈𝑈𝑂

{𝑦𝑟𝑜


∗𝑦𝑟𝑗}

𝜃2�� =2 −

𝑚𝑖𝑛𝑖∈𝐷𝑈𝐼

{ ∑ 𝑛𝑗=1𝜆��

∗∗𝑥𝑖𝑗


𝜁2𝑜∗∗

𝑚𝑎𝑥𝑖∈𝐷𝑈𝐼

{𝑥𝑖𝑜


∗𝑥𝑖𝑗}


1226

is a feasible solution to the model (2-I).

Note that 𝑚𝑎𝑥𝑖∈𝐷𝐷𝐼{

∑ 𝑛𝑗=1𝜆��

∗∗𝑥𝑖𝑗


𝜁1𝑜∗∗ ≤ 1

because for every

𝑖 ∈ 𝐷𝐷𝐼, 𝜁1𝑜∗∗(𝑥𝑖𝑜 + 𝛥𝑥𝑖𝑜) ≥ ∑ 𝑛

𝑗=1 𝜆��∗∗𝑥𝑖𝑗 .

𝜃1��𝑥𝑖𝑜 =2 −


{∑ 𝑛𝑗=1𝜆��

∗∗𝑥𝑖𝑗


𝜁1𝑜∗∗


{𝑥𝑖𝑜


∗𝑥𝑖𝑗}

𝑥𝑖𝑜

≥1


{𝑥𝑖𝑜


∗𝑥𝑖𝑗}𝑥𝑖𝑜

≥1𝑥𝑖𝑜


∗𝑥𝑖𝑗

𝑥𝑖𝑜 =∑ 𝑛𝑗=1 𝜆𝑗

∗𝑥𝑖𝑗

𝑥𝑖𝑜𝑥𝑖𝑜 =∑

𝑛

𝑗=1

𝜆𝑗∗𝑥𝑖𝑗

and, similarly:

𝜃2��𝑥𝑖𝑜 ≤∑

𝑛

𝑗=1

𝜆𝑗∗𝑥𝑖𝑗 ; 𝑖 ∈ 𝐷𝑈𝐼

𝜑1��𝑦𝑟𝑜 ≤∑

𝑛

𝑗=1

𝜆𝑗∗𝑦𝑟𝑗 ; 𝑟 ∈ 𝐷𝑂

𝜑2��𝑦𝑟𝑜 ≥∑

𝑛

𝑗=1

𝜆𝑗∗𝑦𝑟𝑗 ; 𝑟 ∈ 𝑈𝑂.

Clearly, this solution is also satisfied by

the other constraints of model (2-I);

hence, it is a feasible solution to the

model (2-I) thus:

𝜃1𝑜∗ ≤ 𝜃1�� =

2 −𝑚𝑎𝑥𝑖∈𝐷𝐷𝐼

{∑ 𝑛𝑗=1𝜆��

∗∗𝑥𝑖𝑗


𝜁1𝑜∗∗


{𝑥𝑖𝑜


∗𝑥𝑖𝑗}

and consequently:

𝜃1𝑜∗ 𝑚𝑖𝑛𝑖∈𝐷𝐷𝐼

{𝑥𝑖𝑜


∗𝑥𝑖𝑗} ≤ 2 −


{∑ 𝑛𝑗=1𝜆��

∗∗𝑥𝑖𝑗


𝜁1𝑜∗∗


{∑ 𝑛𝑗=1 𝜆��

∗∗𝑥𝑖𝑗


𝜁1𝑜∗∗ ≤ 2 − 𝜃1𝑜

∗ 𝑚𝑖𝑛𝑖∈𝐷𝐷𝐼

{𝑥𝑖𝑜


∗𝑥𝑖𝑗}


{∑ 𝑛𝑗=1𝜆��

∗∗𝑥𝑖𝑗


2 − θ1𝑜∗ 𝑚𝑖𝑛𝑖∈𝐷𝐷𝐼

{xio

∑ nj=1 λj

∗xij}≤ ζ1o

∗∗ .

On the other hand, because λjj=1,...,n∗∗

are

optimal solutions and λjj=1,...,n∗

are feasible

solutions of model(8-I) and the quantity

of slack variables in the optimal solution

must be smaller than those in the feasible

solution, then:

maxi∈DDI

{∑ nj=1 λj

∗∗xij

xio + Δxio} ≥ max

i∈DDI{∑ nj=1 λj

∗xij

xio + Δxio}

and

2 − θ1o∗ mini∈DDI

{xio

∑ nj=1 λj

∗xij} = 1

then:

maxi∈DDI

{∑ nj=1 λj

∗∗xij

xio+Δxio}

2 − θ1o∗ mini∈DDI

{xio

∑ nj=1λj

∗∗xij}≥ max

i∈DDI{∑ nj=1 λj

∗xij

xio + Δxio}

i.e:

ζ1o∗ ≤ ζ1o

∗∗

thus:

ζ1o∗ = ζ1o

∗∗

and therefore the proof is completed.∎

6. Conclusion

In this paper, resource allocation problem

for systems with a supervising center is

investigated. It is assumed that the units

under supervision are homogeneous

because otherwise, the efficiency scores

may reflect the underlying differences in

environments rather than inefficiencies.

In these systems (i.e. organizations)

homogeneous units consume both

desirable and undesirable inputs to

produce outputs some of which might be

undesirable. The proposed method not

only considers environmental factors such

as smoke pollution and waste but also

considers economic ones, both of which

are of fundamental importance to

resource allocation.

As it was discussed, the previous methods

in the literature were not designed to

solve problems with non discretionary

data; while the proposed method , as it

was shown, can do so. Another difference

is that the new method puts emphasis on

efficiency and return to scale

simultaneously as opposed to the previous

IJDEA Vol.4, No.2, (2016).737-749


1227

methods in which resources are allocated

between units to improve efficiency or

maximize the total amount of outputs.

To consider both return to scale and

efficiency simultaneously, our model uses

the notion of MPSS points and allocates

available resources between units in a

way that not only the overall system

distance is minimized, but also the

distance from any unit to it's MPSS points

is minimized as well. In addition to

illustrate the applicability of the proposed

model, two real-life examples were

solved.

The majority of resource allocation

situations are dealt with in the proposed

method. What remains can be of interest

to other scientists and thus subjects for

further researches. Devising a method for

resource allocation in an organization

which deals with fuzzy type data is

proposed as an inspiring further research

question. Moreover, the way data is

handled in this paper might be used as a

guideline for further researches in solving

transportation and/or network problems.


1228

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